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Civil Engineering (CE) Exam  >  Civil Engineering (CE) Tests  >  Test: Gauss's Theorem - Civil Engineering (CE) MCQ

Test: Gauss's Theorem - Civil Engineering (CE) MCQ


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15 Questions MCQ Test - Test: Gauss's Theorem

Test: Gauss's Theorem for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Test: Gauss's Theorem questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Gauss's Theorem MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Gauss's Theorem below.
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Test: Gauss's Theorem - Question 1
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Consider a cube of unit edge length and sides parallel to co-ordinate axes, with its centroid at the point (1, 2, 3). The surface integral  of a vector field  over the entire surface A of the cube is ______.

Detailed Solution for Test: Gauss's Theorem - Question 1

Concept:
Gauss divergence theorem: 

where,   is called surface integral

  is called volume integral

 is called divergence of F

Calculation:

Given:

V = volume of cube = (1)3

= 3 + 5 + 6 = 14

*Answer can only contain numeric values
Test: Gauss's Theorem - Question 2
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If C is the plane triangle which is enclosed by the lines x = π/2,y = 2/πx and y = 0, then the value of following integral ∫C⁡[cos⁡xdy + (y − sin⁡x)dx] comes out to be π/a + b/π. Find |ab|


Detailed Solution for Test: Gauss's Theorem - Question 2

Concept:

Green’s Theorem:

If s is the surface bounded by a closed curve C, then

Calculation:

Given:

f2 = (y – sin x), f1 = cos x

∴ 

∴ a = -4, b = -2

∴ |ab| = |8| = 8

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Test: Gauss's Theorem - Question 3
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The value of ∫S⁡(x + z)dydz + (y + z)dxdz + (x + y)dxdy, Where ‘S’ is the surface of the sphere x2 + y2 + z2 = 4, is ___

Detailed Solution for Test: Gauss's Theorem - Question 3

By Gauss-Divergence Theorem

Test: Gauss's Theorem - Question 4
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If   and s is the closed surface of x2 + y2 + z2 = a2. Then  is
Detailed Solution for Test: Gauss's Theorem - Question 4

Concept:

According to Gauss Divergence theorem:

Calculation:

Given:

x2 + y2 + z2 = a2

∴ closed surface is a sphere of radius r

∴ 

Test: Gauss's Theorem - Question 5
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If S is the surface of the sphere x2 + y2 + z2 = a2, then the value of is
Detailed Solution for Test: Gauss's Theorem - Question 5

Concept:

Gauss Divergence Theorem:

The surface integral of the normal component of a vector function 
 taken around a closed surface S is equal to the integral of the divergence of 
 taken over the volume V enclosed by the surface S. Mathematically;

Calculation:

Given:

 and x2 + y2 + z2 = a2

∴ 

From the Gauss divergence theorem;

where ∫VdV represents volume of sphere.

The equation for the surface of sphere is x2 + y2 + z2 = a2 , ∴  

∴ the radius of sphere is a.

The volume of the sphere = 

⇒ 

⇒ 

Test: Gauss's Theorem - Question 6
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By Gauss divergence theorem the value of   where  and S is  x2 + y2 = 4, z = 0, z = 3 is
Detailed Solution for Test: Gauss's Theorem - Question 6

The given surface is x2 + y2 = 4, z = 0, and z = 3.

Separating the 'y' component of it, we get:

Putting this as a limit in the limits, we get:

= 

= 

= 

= 

= 

 

*Answer can only contain numeric values
Test: Gauss's Theorem - Question 7
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Evaluate the line integral of vector field F = sin y î + x (1 + cos y) ĵ along the circular path given by x2 + y2 = a2, z = 0 (Use a = 5 and round-off to two decimals)

Detailed Solution for Test: Gauss's Theorem - Question 7

Concept:

The position vector 

When z = 0

Circle x2 + y2 = a2 ⇒ x = a cos θ

The line integral is given as:

Calculation:

Now,

x = a cos θ, y = a sin θ

Now:

a = 5 (Given)

∴ 

Test: Gauss's Theorem - Question 8
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Gauss theorem uses which of the following operations?
Detailed Solution for Test: Gauss's Theorem - Question 8

The Gauss divergence theorem uses divergence operator to convert surface to volume integral. It is used to calculate the volume of the function enclosing the region given.

Test: Gauss's Theorem - Question 9
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The Gauss divergence theorem converts
Detailed Solution for Test: Gauss's Theorem - Question 9

The divergence theorem for a function F is given by ∫∫ F.dS = ∫∫∫ Div (F).dV. Thus it converts surface to volume integral.

Test: Gauss's Theorem - Question 10
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Evaluate the surface integral ∫∫ (3x i + 2y j). dS, where S is the sphere given by x2 + y2 + z2 = 9.
Detailed Solution for Test: Gauss's Theorem - Question 10

We could parameterise surface and find surface integral, but it is wise to use divergence theorem to get faster results. The divergence theorem is given by ∫∫ F.dS = ∫∫∫ Div (F).dV
Div (3x i + 2y j) = 3 + 2 = 5.
Now the volume integral will be ∫∫∫ 5.dV, where dV is the volume of the sphere 4πr3/3 and r = 3units.
Thus we get 180π.

Test: Gauss's Theorem - Question 11
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A thin, metallic spherical shell contains a charge Q on it. A point charge q is placed at the centre of the shell and another charge q1 is placed outside it as shown in the following  figure . All the three charges are positive. The force on the central charge due to the shell is
Detailed Solution for Test: Gauss's Theorem - Question 11

We know that electric field lines emerge from a positive charge and move towards a negative charge. Now, charge Q, on the face in front of charge q1, tries to nullify the field lines emerging from charge q1. So, the charge on the farther face of the conductor dominates and, hence, a force appears to the right of charge q.

Test: Gauss's Theorem - Question 12
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A charge Q is uniformly distributed over a large plastic plate. The electric field at a point P close to the centre of the plate is 10 V m−1. If the plastic plate is replaced by a copper plate of the same geometrical dimensions and carrying the same charge Q, the electric field at the point P will become
Detailed Solution for Test: Gauss's Theorem - Question 12

The electric field remains same for the plastic plate and the copper plate, as both are considered to be infinite plane sheets. So, it does not matter whether the plate is conducting or non-conducting.
The electric field due to both the plates,

E = σ/ε0

Test: Gauss's Theorem - Question 13
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Gauss Divergence is defined as:
Detailed Solution for Test: Gauss's Theorem - Question 13

Gauss Divergence theorem:

Green's theorem:

Stokes theorem:

Test: Gauss's Theorem - Question 14
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The value of the integral over the closed surface S bounding a volume V, where  is the position vector and n̂ is normal to the surface S, is
Detailed Solution for Test: Gauss's Theorem - Question 14

Concept:

According to Gauss Divergence theorem:

Calculation:

Given:

that S is a closed surface:

Test: Gauss's Theorem - Question 15
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Gauss Divergence is defined as:
Detailed Solution for Test: Gauss's Theorem - Question 15

Gauss Divergence theorem:

Green's theorem:

Stokes theorem:

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